Collaborative Research: Asymptotic Geometry and Analysis of Stochastic Partial Differential Equations

这个项目涉及发展随机场的几何/解析理论，主要是那些产生于随机偏微分方程组[简称SPDEs]的理论。特别强调在纯数学和应用数学、数学海洋学、随机水文学、地质统计学和数学物理的各个领域中发挥中心作用的某些SPDEs和相关的随机场。研究人员将开发概率、几何和分析工具，这些工具将导致对一大类物理和数学上感兴趣的SPDEs和相关随机场有更深入的了解。调查人员认为，这些工具将具有足够的新颖性，开辟新的研究领域，解决SPDEs理论和相关随机领域的一些长期悬而未决的问题，并进一步促进其适用性。此外，这些活动包括培养研究生和博士后学者的持续计划，并在数学和统计科学领域发展他们的职业生涯。表征SPDEs及其相关随机场的精细局部和渐近结构既具有重要意义，又具有挑战性。在过去的研究中，研究者们已经建立了一系列关于SPDEs及其相关随机场解的渐近性、间歇性和宏观尺度多重分形性质的结果，发展了关于加性Levy过程和Brown Sheet的势理论，并利用它们解决了Levy过程、Brown Sheet和抛物型随机偏微分方程理论中的几个未决问题。研究人员以概率论和几何测度论为基础，发展了分析非马尔可夫高斯和稳定随机场的思想。他们还引入了更新理论和耦合技术来对一大类非线性单项微分方程解的渐近分析。他们计划继续研究SPDEs、随机场、位势理论和随机分形几何之间的精确定量联系。他们相信，对这些联系的进一步追求最终将产生对SPDEs、大规模物理多重分形图和相关随机场的结构的新见解。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project is concerned with the development of a geometric/analytic theory of random fields, primarily those that arise from stochastic partial differential equations [SPDEs, for short]. Special emphasis is placed on certain SPDEs and related random fields that play a central role in various areas of pure and applied mathematics, mathematical oceanography, stochastic hydrology, geostatistics, and mathematical physics. The investigators will develop probabilistic, geometric, and analytic tools that will lead to a deeper understanding of a large family of physically- and mathematically-interesting SPDEs and related random fields. The investigators believe that these tools will have sufficient novelty to open new research areas, solve a number of long-standing open problems in the theory of SPDEs and related random fields, and also further promote their applicability. In addition, the activities include a sustained program to train graduate students and postdoctoral scholars, and to develop their careers in the mathematical and statistical sciences.It is both significant and challenging to characterize the fine local and asymptotic structure of SPDEs and related random fields. In their past investigations, the investigators have established a series of results on the asymptotic behavior, intermittency, and macroscopic-scale multifractal properties of the solutions of SPDEs and related random fields, developed potential theories for additive Levy processes and the Brownian sheet, and used them to resolve several outstanding open problems in Levy processes, the Brownian sheet, and the theory of parabolic stochastic partial differential equations. The investigators have developed ideas, based on probability theory and geometric measure theory, for the analysis of non-Markovian Gaussian and stable random fields. They have also introduced renewal-theoretic and coupling techniques for the asymptotic analysis of solutions to a large class of nonlinear SPDEs. They plan to continue their investigation of precise quantitative connections between SPDEs, random fields, potential theory, and the geometry of random fractals. They believe that further pursuit of these connections will ultimately yield novel insights into the structure of SPDEs,large-scale physical multifractals, and related random fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.